3.1: One-Step Equations
Learning Objectives
At the end of this lesson, students will be able to:
- Solve an equation using addition.
- Solve an equation using subtraction.
- Solve an equation using multiplication.
- Solve an equation using division.
Vocabulary
Terms introduced in this lesson:
- equal
- equation
- isolate
- linear equation
Teaching Strategies and Tips
Use the introductory problem to motivate equivalent equations, since it can be solved two ways:
- The cost plus the change received is equal to the amount paid, \begin{align*}x + 22 = 100\end{align*}.
- The cost is equal to the difference between the amount paid and the change, \begin{align*}x = 100 - 22\end{align*}
Examples 1 and 2 are essentially the same; constant terms must be added to both sides to isolate \begin{align*}x\end{align*} on the left. Adding vertically can benefit students.
Additional Example.
Solve \begin{align*}12 = -4 + x\end{align*}.
Solution. To isolate \begin{align*}x\end{align*}, add \begin{align*}4\end{align*} to both sides of the equation. Add vertically.
\begin{align*}12 & = -\cancel{4} + x\\ +4 & = +\cancel{4}\\ 16 & = x\end{align*}
The variable in Example 3 is not the usual \begin{align*}x\end{align*}. Remind students that the letter of the variable does not matter. Additional Example.
Solve \begin{align*}-21=n+14\end{align*}.
Hint: To isolate \begin{align*}n\end{align*}, subtract \begin{align*}14\end{align*} from both sides of the equation.
In Examples 4-6, teachers may opt to solve the equations by adding the opposite in lieu of subtracting.
Example.
Solve \begin{align*}-17=x+8\end{align*}.
Hint: To isolate \begin{align*}x\end{align*}, add \begin{align*}-8\end{align*} to both sides of the equation. Add vertically.
\begin{align*}-17 & = x+\cancel{8}\\ -8 & = -\cancel{8}\\ -25 & = x\end{align*}
Use Example 6 as an example of an equation with fractions. Remind students to find common denominators.
Point out in Example 8 that in general,
\begin{align*}\frac{ax}{b}=\frac{a}{b}x\end{align*}
which will help students isolate \begin{align*}x\end{align*} in one step, multiplying by the reciprocal of \begin{align*}a/b\end{align*}.
Note that the equation in Example 10 can be written in dollars or in cents:
- \begin{align*}5x=3.25\end{align*} (\begin{align*}x\end{align*} in dollars)
- \begin{align*}5x=325\end{align*} (\begin{align*}x\end{align*} in cents)
Although Examples 13 and 15 can be solved by making a table and Example 14 by guessing and checking, teachers are encouraged to help students setup and solve an equation of the type presented in the lesson.
Error Troubleshooting
General Tip: After the constant term is canceled in a one-step equation, the variable must be carried down onto the next line. Remind students to write the \begin{align*}x =\end{align*}.
General Tip: Students forget to perform the same operation on both sides of an equation. Have students use a colored pencil to write what they are doing to both sides of the equation.
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